In Munich - I always enjoyed walking around Marienplatz and those areas just because its pleasant and seeing markets (it's been a decade since I've been). I think there is some cool Bavarian castle outside of town as well.

Obviously there is beer. I would not go to Hofbrauhaus since you can get that here and around the world. Maybe hit the other ones, Augustiner (sp), Im Tal and so many others. It's a cool way to start an evening because the beers are huge, delicious and should be reasonably priced. Lots of families and friend groups around so its very casual and welcoming. They sell their own food, but I think you can bring your own food as well picnic style. Might be a nice way to meet some friendly locals. Of course sausage (heh heh heh) is big in Munich. So an old school breakfast is Weisswurst (white sausage) with Hefeweisen beer. It felt like a weird combination at 8am (on my quick layover in Munich) but was delicious. Leberkeisen (sp?) is also delicious. Remember people in Munich are Bavarian (and only after that, German) and take great pride in that. Of course like in much of the rest of the country the bread/pastries etc will be delicious. Didn't get to party so can't help you there. The interwebs can though.

Copenhagen has been covered by someone else. Some great restaurants and museums and just a cool city in general. I think I was drunk all 4 days I was there and so the nightlife can be expensive but lots of fun. I was with a local buddy so can't help there.

Vienna I actually found quite boring, sadly. Not a ton going on. Maybe some ok museums or art, but not much else (maybe I'm wrong and I would love to be corrected). But pleasant. That said the Melange (a Viennese form of coffee) and Sachertorte (a Viennese cake) are delicious and well worth it.

If you have that many days between both Munich and Vienna I highly suggest taking a train / day trip to Salzburg. One of the most beautiful places I've ever visited. You can also get the Sachertorte there instead of Vienna (with much less of a line/wait and it'll be a few euro cheaper). I've visited Austria on 4 separate Europe trips and if I hit Vienna or Salzburg I always make it a point to get a slice. In Vienna I'd you're into politics the UN building was really interesting - book a private tour. Also go see an Opera. Even if it's not your thing (its not mine), it's truly an unforgettable experience and will give you a new appreciation for the arts.

In Munich you should walk up the belfry (it's a few hundred winding and narrow stairs so wear comfortable shoes) starting around 1130am so you can be up top at noon when the glockenspiel on the Altes Rathaus plays - it's a really cool thing to see and hear. I'd disagree with the other poster and suggest going to Hofbrauhaus, in addition to other beer halls (definitely do Augustiner). Also make sure to eat some pork knuckle while in the city.